A strongly correlated metal built from Sachdev-Ye-Kitaev models

TL;DR

This work exemplifies an analytically controlled study of a strongly correlated metal crossover in full detail, including thermodynamics, low temperature Landau quasiparticle interactions, and both electrical and thermal conductivity at all scales.

Abstract

Strongly correlated metals comprise an enduring puzzle at the heart of condensed matter physics. Commonly a highly renormalized heavy Fermi liquid occurs below a small coherence scale, while at higher temperatures a broad incoherent regime pertains in which quasi-particle description fails. Despite the ubiquity of this phenomenology, strong correlations and quantum fluctuations make it challenging to study. The Sachdev-Ye-Kitaev(SYK) model describes a $0+1$D quantum cluster with random all-to-all \emph{four}-fermion interactions among $N$ Fermion modes which becomes exactly solvable as $N\rightarrow \infty$, exhibiting a zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absence of quasi-particles. Here we study a lattice of complex-fermion SYK dots with random inter-site \emph{quadratic} hopping. Combining the imaginary time path integral with \emph{real} time path integral formulation, we obtain a heavy Fermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperature Landau quasiparticle interactions, and both electrical and thermal conductivity at all scales. We find linear in temperature resistivity in the incoherent regime, and a Lorentz ratio $L\equiv \frac{κρ}{T}$ varies between two universal values as a function of temperature. Our work exemplifies an analytically controlled study of a strongly correlated metal.

Figures (4)

• Figure 1: The entropy and specific heat(inset) collapse to universal functions of $\frac{T}{E_c}$, given $t_0,T\ll U_0$($z=2$). $C\rightarrow \mathcal{S}'(0)T/E_c$ as $T/E_c\rightarrow 0$. Solid curves are guides to the eyes.
• Figure 2: The spectral weight $A(\omega)$ at fixed $U_0/T=10^4,\mu=0,z=2$ for $E_c/T=0,0.09,1,9$, corresponding a crossover from SYK$_4$ limit to the "heavy Fermi liquid" regime. Inset shows the comparison of green's function for $T/E_c=9$ with free fermion limit result.
• Figure 3: (a): For $t_0, T\ll U_0$, $\rho_{\varphi/\varepsilon}$ "collapse" to $R_{\varphi/\varepsilon}(\frac{T}{E_c})/N$. (b): The Lorentz ratio $\frac{\kappa\rho}{T}$ reaches two constants $\frac{\pi^2}{3},\frac{\pi^2}{8}$, in the two regimes. The solid curves are guides to the eyes.
• Figure 4: $t$ in the legend denotes $t=\sqrt{\frac{z}{2}} t_0$ where $z$ is the coordination number. Left: The entropy for $\text{SYK}_4$ (i.e. vanishing $t_0$) agrees with the results in Ref fu_2016 and entropy approaches identically regardless of $t/U_0$ the universal $\ln 2$ for high temperature (not shown in the figure). For nonzero $t$, entropy tends to vanish identically as temperature $T\rightarrow 0$. Right: The compressibility $K$ extracted by virtue of $K=\frac{1}{N}\frac{\partial \mathcal{N}}{\partial \mu}$. For $t/U_0\ll 1$ and relatively small temperature (i.e., $T\ll U_0$), $K$ is independent of temperature and varies little with $t$.